Q: Which of the following is a correct trigonometric identity?

$$\frac{1+\sin x}{1-\sin x} = (\tan x + \sec x)^2$

Question 1

A Ferris wheel has a radius of 18 m and its center is 22 m above the ground. One complete revolution takes 40 seconds. A rider starts at the lowest point at $$t = 0. $$What is the equation for the rider's height $$h(t)$$ above the ground after $$t$$ seconds?

Accepted Answer

$$h(t) = 18 \cos\left(\frac{\pi}{20} t + \pi\right) + 22$$

Question 2

An arc of length 100 m subtends a central angle in a circle of radius 50 m. What is the measure of the central angle in radians?

Accepted Answer

$$2$$ radians

Question 3

Convert the central angle from the previous question to degrees.

Accepted Answer

$$114.6^\circ$$

Question 4

A 20-foot ladder is leaning against a building. The base of the ladder is 6 feet from the building. What is the angle of elevation $$	heta of $$the ladder (to one decimal place)?

Accepted Answer

$$72.5^\circ$$

Question 5

Given $$	an 	heta = -\frac{1}{4}$$ and $$\sec 	heta \u003c 0$$, in which quadrant does $$	heta$$ lie?

Accepted Answer

Quadrant II

Question 6

Given $$	an 	heta = -\frac{1}{4}$$ and $$\sec 	heta \u003c 0$$, what is $$\sin 	heta$$?

Accepted Answer

$$-\frac{1}{\sqrt{17}}$$

Question 7

Find the exact value of $$	an\left(\sin$$^{-1}$$\left(-\frac{\sqrt{3}}{3}ight)ight).$$

Accepted Answer

$$-\frac{\sqrt{3}}{3}$$

Question 8

Given triangle $$ABC$$ with $$a = 8$$, $$b = 11$$, and $$c = 14$$, which law would you use first to solve for the angles?

Accepted Answer

Law of Cosines

Question 9

If $$	heta = \frac{\pi}{3}$$, what is $$\sin\left(\frac{1}{2}	hetaight)$$?

Accepted Answer

$$\frac{1}{2}$$

Question 10

Are $$\sin\left(\frac{1}{2}	hetaight)$$ and $$\frac{1}{2}\sin 	heta$$ the same for $$	heta = \frac{\pi}{3}$$?

Accepted Answer

No, they are not equal.

Question 11

Write $$\cos\left(\sin$$^{-1}$$ x - 	an$$^{-1}$$ yight) in $$terms of $$x$$ and $$y$$ only.

Accepted Answer

$$\sqrt{1-x^2$} \cdot \frac{1}{\sqrt{1+y^2$}} + x \cdot \frac{y}{\sqrt{1+y^2$}}$$

Question 12

Express $$\sin\left(2 	an$$^{-1}$$ xight) as an $$algebraic expression in $$x.$$

Accepted Answer

$$\frac{2x}{1+x^2$}$$

Question 13

Which of the following is a correct trigonometric identity?

Accepted Answer

$$\frac{1+\sin x}{1-\sin x} = (	an x + \sec x)^2$

Question 14

Solve for $$x in$$ $$0 \leq x \u003c 2\pi$$: $$\csc^2$ x - 4 = 0.$$

Accepted Answer

$$x = \frac{\pi}{2}, \frac{3\pi}{2}$$

Question 15

Find all solutions in $$0 \leq x \u003c 2\pi$$ for $$2 \sin^2$ x + 5 \sin x - 12 = 0.$$

Accepted Answer

$$x = \frac{\pi}{2}, \frac{7\pi}{6}, \frac{11\pi}{6}$$

Question 16

Solve $$2 \cos(3x) + 1 = 0$$ for $$0 \leq x \u003c 2\pi.$$

Accepted Answer

$$x = \frac{\pi}{9}, \frac{5\pi}{9}, \frac{13\pi}{9}$$

Question 17

A sector of a circle has a radius of 50 m and a central angle of 2 radians. What is the area of the sector?

Accepted Answer

$$2500 m$^2$$

Question 18

If $$2 \sin^2$ x = 2 + \cos(2x)$$, what is a solution for $$x in$$ $$[0, 2\pi)$$?

Accepted Answer

$$x = \frac{\pi}{2}$$

Precalculus Practice: Trigonometry, Circles, and Triangle Solving

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